This invention relates for obtaining and computing the subsurface temperature depth distribution along with its error bounds. The solution has been determined for the stochastic heat conduction equation by considering different sets of boundary conditions and radiogenic heat sources and incorporating randomness in the thermal conductivity. In understanding the Earth thermal structure there are several questions which need clear answers. Many of the controlling parameters that define the Earth's processes are not known with certainty. In such situations these controlling parameters can be defined in a stochastic framework and an average picture of the system behavior together with its error bounds can be quantified.
The thermal structure of the Earth's crust is influenced by its geothermal parameters such as thermal conductivity, radiogenic heat sources and initial and boundary conditions. Basically two approaches of modeling are commonly used for the estimation of the subsurface temperature field. These are: (1) deterministic approach and (2) the stochastic approach. In the deterministic approach the subsurface temperature field is obtained assuming that the controlling thermal parameters are known with certainty. However, due to inhomogeneous nature of the Earth's interior some amount of uncertainty in the estimation of the geothermal parameters are bound to exist. Uncertainties in these parameters may arise from the inaccuracy of measurements or lack of information about the parameters themselves. Such uncertainties in parameters are incorporated in the stochastic approach and an average picture of the thermal field along with its associated error bounds are determined. To assess the properties of the system at a glance we need to obtain the mean value that gives the average picture and the variance or the standard deviation that is the variability indicator which gives the errors associated with the system behavior due to errors in the system input.
Subsurface temperatures are also seen to be very sensitive to perturbations in the input thermal parameters and hence several studies have been carried out in quantifying the perturbations in the temperatures and heat flow using stochastic analytical and random simulation techniques. Quantification of uncertainty in the heat flow using a least squares inversion technique incorporating uncertainties in the temperature and thermal conductivities has been done, Tectonophysics, Vol 121, 1985 by Vausser et al. The effect of variation in heat source on the surface heat flow has also been studied, Journal Geophysical Research, V 91, 1986, by Vasseur and Singh, Geophysical Research Letters, V14, 1987, by Nielsen. In most of the studies the stochastic heat equation has been solved using the small perturbation method. Using the small perturbation method the heat conduction equation has been solved by incorporating uncertainties in the heat sources and the mean temperature field along with its error bounds have been obtained, Geophysical Journal International, 135, 1998, by Srivastava and Singh. The random simulation method has also been used to model the thermal structure incorporating uncertainties in the controlling thermal parameters, Tectonophysics, V156, 1988 by Royer and Danis, Marine and Petroleum Geology, V 14, 1997, by Gallagher et al, Tectonophysics, V 306, 1999a, b, by Jokinen and Kukkonen. This numerical modeling is very useful in studying the nonlinear problems but sometimes simple 1-D analytical solution to the mean behavior and its associated error bounds is very useful in quantifying the uncertainty. The stochastic differential equations in other fields are now being solved by yet another approach called the decomposition method, Journal of Hydrology, V 169, 1995, by Serrano. In a recent study using this new approach the stochastic heat equation has been solved incorporating uncertainties in the thermal conductivity where the solution to the temperature field is obtained using a series expansion method, Geophysical Journal International, V 138, 1999, by Srivastava and Singh. The thermal conductivity is considered to be a random parameter with a known Gaussian colored noise correlation structure.
In this invention the stochastic solution to the mean and variance in the temperature field for a different set of boundary conditions and different radiogenic heat source function has been obtained following the procedure of Geophysical Journal International, V 138, 1999, by Srivastava and Singh. The expressions for mean and variance in temperature depth distribution for different heat sources and boundary conditions have been obtained and used to compute and plot the subsurface thermal field along with its error bounds.